Obviously, as any Anglican priest could point out for them, it's not so much a matter of either/or, as both/and. Jesus could probably actually speak or at least understand several languages, to different degrees:
- Maybe the odd word of Latin
- Somewhere behind a few words and a working, if limited, Koine Greek
- Aramaic as his conversational language, at home or with friends
- The Hebrew of the Old Testament in the Synagogue.
And of course, when speaking "On the Record", as it were, when he was going to be quoted by the Gospels, he spoke Jacobean English.